Spot-Futures Parity

Arbitrage and spot-futures parity

• In last session, looked at implications of the expectations hypothesis for the forward curve.
• In this session, we look at the implications of arbitrage.
• Question is: how do futures prices relate to spot (contango or backwardation) and why?
• Link between spot and futures is called spot-futures parity.

Overview of arbitrage

• Suppose the forward curve is extremely steep. At some point, it pays to buy spot and sell futures.
• Suppose the forward curve is extremely downwards sloping.
  • If you can short sell the spot, you can sell spot and buy futures. Accept delivery on the futures to cover the short spot position.
  • Or people who are long spot, could sell it and buy futures to restore their long position.

Synthetic long futures and steep forward curves

• On a long futures (ignoring daily settlement), you pay at the delivery date and accept delivery then.
• To duplicate this, you can borrow money, buy spot and hold until the delivery date.
• At the delivery date, you will have the asset and owe money (so you pay at the delivery date).
• If the synthetic futures is cheaper, you should create it and sell the futures contract.

• The synthetic futures is cheaper when the futures price is sufficiently high relative to spot (forward curve is sufficiently steep).
• So, arbitrage rules out extremely steep forward curves.

Synthetic short futures and downward sloping forward curves

• On a short futures (ignoring daily settlement), you deliver at the delivery date and get paid then.
• To duplicate this, you can short the spot and invest the proceeds in T-bills.
• At the delivery date, you cover the short and have cash (T-bills).
• If the synthetic short futures generates more cash, then you should create it and buy the futures contract.

• The synthetic short futures generates more cash when the spot price is high relative to the futures price.
• So, arbitrage rules out extremely downward sloping forward curves (when you can short the asset).

Cost of carry and convenience yield

• Now, more details.
  
• On the synthetic long futures, you pay at delivery the spot price plus interest.
• Also, there may be storage costs.
• Cost of carry = interest + storage costs.
• On the other hand, you may earn dividends, etc. when holding the asset.
• Convenience yield = dividends or other income

• Ruling out extremely steep forward curves means

$$ +

- $$

• Ruling out extremely downward sloping forward curves (when you can short) means

\[ \text{futures price} \ge \text{spot price} + \text{cost of carry} - \text{convenience yield}\]

Gold

• Convenience yield = 0
• Storage costs \(\approx\) 0
• Maybe cannot short, but lots of gold in storage (longs can sell and then restore, like shorting and covering)
• So,

Understanding S&P 500 Futures

• Buying stocks ties up money but provides dividends.
• Better to buy spot (at equal prices) if dividends > interest. This implies backwardation.
• Better to buy futures (at equal prices) if interest > dividends. This implies contango.

Currency Futures

• The advantage of owning a foreign currency now is that you can invest it and earn the foreign interest rate.

• The cost of buying foreign currency now is that it ties up dollars and you lose the dollar interest rate.

• Whether it is better to buy now or buy later depends on which interest rate is higher.

• U.S. interest rate higher \(\Rightarrow\) contango (like gold)

• Foreign interest rate higher \(\Rightarrow\) backwardation.

Euro Forward Curve

Euro forward curve on 1-24-2022

Peso Forward Curve

Mexican peso forward curve on 1-24-2022

Commodities

• Commodities (energy, agricultural) are sometimes in backwardation and sometimes in contango.

• Backwardation occurs

  • when there is nothing left in storage to sell now to cause spot prices to come down relative to futures prices.
  • or people are hoarding. For example, corn (and others) moved into backwardation during the pandemic.

• If a commodity is seasonal and storage is costly, futures may have seasonal spikes (e.g., natural gas and gasoline).

Crude Oil Forward Curves

Backwardation and contango in 2020

Natural Gas

Natural gas forward curve on 1-24-2022

Gasoline

Gasoline forward curve on 1-24-2022

Understanding Gold Futures

• There is lots of gold in storage. If the futures is less than the spot price, people storing gold should sell spot and buy futures.
  • They earn the difference between the prices plus interest that they can earn on the cash from selling spot.
• The story is the same if the futures price at a long maturity is less than the futures price at a shorter maturity.
  • Buy the cheap short maturity and sell the expensive long maturity.

• But what everyone thinks the bottom will drop out of the gold market?

• Then the spot price will drop now. The futures will still end up higher than the spot.

Gold futures

• Synthetic gold futures:
  • borrow money, buy gold spot and store it
  • later pay today’s spot price + interest and have gold
• Futures price should equal today’s spot price + interest
  • Anyone can create synthetic futures and sell actual futures if synthetic futures is cheaper
  • People storing gold can “sell” synthetic futures and buy actual if synthetic is more expensive

Cost of carry and convenience yield

• Cost of carry = cost of owning physical = foregone interest + storage costs
• Convenience yield = benefit of owning physical = dividends and/or profits from temporary product shortages

Contango and backwardation

• Contango = upward sloping forward curve (futures > spot)
• Cost of carry > convenience yield \(\Rightarrow\) contango (like gold)
• Backwardation = downward sloping (spot > futures)
• Convenience yield > cost of carry \(\Rightarrow\) backwardation (like peso)
• Commodities can have high cost of carry (storage) and sometimes high convenience yield